1% generated by GAPDoc2LaTeX from XML source (Frank Luebeck) 2\documentclass[a4paper,11pt]{report} 3 4\usepackage[top=37mm,bottom=37mm,left=27mm,right=27mm]{geometry} 5\sloppy 6\pagestyle{myheadings} 7\usepackage{amssymb} 8\usepackage[latin1]{inputenc} 9\usepackage{makeidx} 10\makeindex 11\usepackage{color} 12\definecolor{FireBrick}{rgb}{0.5812,0.0074,0.0083} 13\definecolor{RoyalBlue}{rgb}{0.0236,0.0894,0.6179} 14\definecolor{RoyalGreen}{rgb}{0.0236,0.6179,0.0894} 15\definecolor{RoyalRed}{rgb}{0.6179,0.0236,0.0894} 16\definecolor{LightBlue}{rgb}{0.8544,0.9511,1.0000} 17\definecolor{Black}{rgb}{0.0,0.0,0.0} 18 19\definecolor{linkColor}{rgb}{0.0,0.0,0.554} 20\definecolor{citeColor}{rgb}{0.0,0.0,0.554} 21\definecolor{fileColor}{rgb}{0.0,0.0,0.554} 22\definecolor{urlColor}{rgb}{0.0,0.0,0.554} 23\definecolor{promptColor}{rgb}{0.0,0.0,0.589} 24\definecolor{brkpromptColor}{rgb}{0.589,0.0,0.0} 25\definecolor{gapinputColor}{rgb}{0.589,0.0,0.0} 26\definecolor{gapoutputColor}{rgb}{0.0,0.0,0.0} 27 28%% for a long time these were red and blue by default, 29%% now black, but keep variables to overwrite 30\definecolor{FuncColor}{rgb}{0.0,0.0,0.0} 31%% strange name because of pdflatex bug: 32\definecolor{Chapter }{rgb}{0.0,0.0,0.0} 33\definecolor{DarkOlive}{rgb}{0.1047,0.2412,0.0064} 34 35 36\usepackage{fancyvrb} 37 38\usepackage{mathptmx,helvet} 39\usepackage[T1]{fontenc} 40\usepackage{textcomp} 41 42 43\usepackage[ 44 pdftex=true, 45 bookmarks=true, 46 a4paper=true, 47 pdftitle={Written with GAPDoc}, 48 pdfcreator={LaTeX with hyperref package / GAPDoc}, 49 colorlinks=true, 50 backref=page, 51 breaklinks=true, 52 linkcolor=linkColor, 53 citecolor=citeColor, 54 filecolor=fileColor, 55 urlcolor=urlColor, 56 pdfpagemode={UseNone}, 57 ]{hyperref} 58 59\newcommand{\maintitlesize}{\fontsize{50}{55}\selectfont} 60 61% write page numbers to a .pnr log file for online help 62\newwrite\pagenrlog 63\immediate\openout\pagenrlog =\jobname.pnr 64\immediate\write\pagenrlog{PAGENRS := [} 65\newcommand{\logpage}[1]{\protect\write\pagenrlog{#1, \thepage,}} 66%% were never documented, give conflicts with some additional packages 67 68\newcommand{\GAP}{\textsf{GAP}} 69 70%% nicer description environments, allows long labels 71\usepackage{enumitem} 72\setdescription{style=nextline} 73 74%% depth of toc 75\setcounter{tocdepth}{1} 76 77 78 79 80 81%% command for ColorPrompt style examples 82\newcommand{\gapprompt}[1]{\color{promptColor}{\bfseries #1}} 83\newcommand{\gapbrkprompt}[1]{\color{brkpromptColor}{\bfseries #1}} 84\newcommand{\gapinput}[1]{\color{gapinputColor}{#1}} 85 86 87\begin{document} 88 89\logpage{[ 0, 0, 0 ]} 90\begin{titlepage} 91\mbox{}\vfill 92 93\begin{center}{\maintitlesize \textbf{NoCK\mbox{}}}\\ 94\vfill 95 96\hypersetup{pdftitle=NoCK} 97\markright{\scriptsize \mbox{}\hfill NoCK \hfill\mbox{}} 98{\Huge \textbf{Computing obstruction for the existence of compact Clifford-Klein form\mbox{}}}\\ 99\vfill 100 101{\Huge Version 1.4\mbox{}}\\[1cm] 102{October 2019 \mbox{}}\\[1cm] 103\mbox{}\\[2cm] 104{\Large \textbf{ Maciej Boche{\a'n}ski \mbox{}}}\\ 105{\Large \textbf{ Piotr Jastrz{\k e}bski \mbox{}}}\\ 106{\Large \textbf{ Anna Szczepkowska \mbox{}}}\\ 107{\Large \textbf{ Aleksy Tralle \mbox{}}}\\ 108{\Large \textbf{ Artur Woike \mbox{}}}\\ 109\hypersetup{pdfauthor= Maciej Boche{\a'n}ski ; Piotr Jastrz{\k e}bski ; Anna Szczepkowska ; Aleksy Tralle ; Artur Woike } 110\end{center}\vfill 111 112\mbox{}\\ 113{\mbox{}\\ 114\small \noindent \textbf{ Maciej Boche{\a'n}ski } Email: \href{mailto://mabo@matman.uwm.edu.pl} {\texttt{mabo@matman.uwm.edu.pl}}\\ 115 Address: \begin{minipage}[t]{8cm}\noindent 116 Faculty of Mathematics and Computer Science,\\ 117 University of Warmia and Mazury in Olsztyn\\ 118 Sloneczna 54 Street, \\ 119 10-710 Olsztyn, Poland \end{minipage} 120}\\ 121{\mbox{}\\ 122\small \noindent \textbf{ Piotr Jastrz{\k e}bski } Email: \href{mailto://piojas@matman.uwm.edu.pl} {\texttt{piojas@matman.uwm.edu.pl}}\\ 123 Homepage: \href{http://wmii.uwm.edu.pl/~piojas/} {\texttt{http://wmii.uwm.edu.pl/\texttt{\symbol{126}}piojas/}}\\ 124 Address: \begin{minipage}[t]{8cm}\noindent 125 Faculty of Mathematics and Computer Science,\\ 126 University of Warmia and Mazury in Olsztyn\\ 127 Sloneczna 54 Street, \\ 128 10-710 Olsztyn, Poland \end{minipage} 129}\\ 130{\mbox{}\\ 131\small \noindent \textbf{ Anna Szczepkowska } Email: \href{mailto://anna.szczepkowska@matman.uwm.edu.pl} {\texttt{anna.szczepkowska@matman.uwm.edu.pl}}\\ 132 Address: \begin{minipage}[t]{8cm}\noindent 133 Faculty of Mathematics and Computer Science,\\ 134 University of Warmia and Mazury in Olsztyn\\ 135 Sloneczna 54 Street, \\ 136 10-710 Olsztyn, Poland \end{minipage} 137}\\ 138{\mbox{}\\ 139\small \noindent \textbf{ Aleksy Tralle } Email: \href{mailto://tralle@matman.uwm.edu.pl} {\texttt{tralle@matman.uwm.edu.pl}}\\ 140 Homepage: \href{http://wmii.uwm.edu.pl/~tralle/} {\texttt{http://wmii.uwm.edu.pl/\texttt{\symbol{126}}tralle/}}\\ 141 Address: \begin{minipage}[t]{8cm}\noindent 142 Faculty of Mathematics and Computer Science,\\ 143 University of Warmia and Mazury in Olsztyn\\ 144 Sloneczna 54 Street, \\ 145 10-710 Olsztyn, Poland \end{minipage} 146}\\ 147{\mbox{}\\ 148\small \noindent \textbf{ Artur Woike } Email: \href{mailto://awoike@matman.uwm.edu.pl} {\texttt{awoike@matman.uwm.edu.pl}}\\ 149 Homepage: \href{http://wmii.uwm.edu.pl/~awoike/} {\texttt{http://wmii.uwm.edu.pl/\texttt{\symbol{126}}awoike/}}\\ 150 Address: \begin{minipage}[t]{8cm}\noindent 151 Faculty of Mathematics and Computer Science,\\ 152 University of Warmia and Mazury in Olsztyn\\ 153 Sloneczna 54 Street, \\ 154 10-710 Olsztyn, Poland \end{minipage} 155}\\ 156\end{titlepage} 157 158\newpage\setcounter{page}{2} 159{\small 160\section*{Abstract} 161\logpage{[ 0, 0, 1 ]} 162 In this package we develop functions for an algorithm designed to find 163homogeneous spaces of semisimple non-compact Lie groups which do not admit 164compact Clifford-Klein forms. \mbox{}}\\[1cm] 165{\small 166\section*{Copyright} 167\logpage{[ 0, 0, 3 ]} 168 NoCK Package is free software; you can redistribute it and/or modify it under 169the terms of the \href{http://www.fsf.org/licenses/gpl.html} {GNU General Public License} as published by the Free Software Foundation; either version 2 of the License, 170or (at your option) any later version. \mbox{}}\\[1cm] 171{\small 172\section*{Acknowledgements} 173\logpage{[ 0, 0, 2 ]} 174 We thank Willem de Graaf for his help in getting some literature sources. \mbox{}}\\[1cm] 175\newpage 176 177\def\contentsname{Contents\logpage{[ 0, 0, 4 ]}} 178 179\tableofcontents 180\newpage 181 182 183\chapter{\textcolor{Chapter }{Notation}}\logpage{[ 1, 0, 0 ]} 184\hyperdef{L}{X7DD31B407B9402A3}{} 185{ 186 We use the notation and convention for real Lie algebras as is from CoReLG 187Package, \cite{CoReLG}. 188\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] 189 !gapprompt@gap>| !gapinput@G:=RealFormById( "E", 7,3); 190| 191 <Lie algebra of dimension 133 over SqrtField> 192 !gapprompt@gap>| !gapinput@rankG:=Dimension(CartanSubalgebra(G)); 193| 194 7 195 !gapprompt@gap>| !gapinput@rankRG:=Dimension(CartanSubspace(G)); 196| 197 3 198 !gapprompt@gap>| !gapinput@dimG:=Dimension(G); 199| 200 133 201 !gapprompt@gap>| !gapinput@P:=CartanDecomposition( G ).P; 202| 203 <vector space over SqrtField, with 54 generators> 204 !gapprompt@gap>| !gapinput@dimPforG:=Dimension(P); 205| 206 54 207 !gapprompt@gap>| !gapinput@K:=CartanDecomposition( G ).K; 208| 209 <Lie algebra of dimension 79 over SqrtField> 210 !gapprompt@gap>| !gapinput@rankK:= Dimension(CartanSubalgebra(K)); 211| 212 7 213 !gapprompt@gap>| !gapinput@dimK:= Dimension(K); 214| 215 79 216\end{Verbatim} 217 Classification can be found in Table 9 in \cite{onvin}, p. 312-317. } 218 219 220\chapter{\textcolor{Chapter }{Obstruction for the existence of compact Clifford-Klein form}}\logpage{[ 2, 0, 0 ]} 221\hyperdef{L}{X7DD2840E797415E3}{} 222{ 223 In this chapter we describe functions for algorithm from \cite{our}. 224\section{\textcolor{Chapter }{Technical functions}}\logpage{[ 2, 1, 0 ]} 225\hyperdef{L}{X805C42557996F58A}{} 226{ 227 228 229\subsection{\textcolor{Chapter }{NonCompactDimension}} 230\logpage{[ 2, 1, 1 ]}\nobreak 231\hyperdef{L}{X832D9E887973AABE}{} 232{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{NonCompactDimension({\mdseries\slshape G})\index{NonCompactDimension@\texttt{NonCompactDimension}} 233\label{NonCompactDimension} 234}\hfill{\scriptsize (function)}}\\ 235 236 237 For a real Lie algebra $G$ constructed by the function \mbox{\texttt{\mdseries\slshape RealFormById}} (from \cite{CoReLG}), this function returns the non-compact dimension of $G$ (dimension of a non-compact part in Cartan decomposition of $G$). 238\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] 239 !gapprompt@gap>| !gapinput@G:=RealFormById("E",6,2); # E6(6) 240| 241 <Lie algebra of dimension 78 over SqrtField> 242 !gapprompt@gap>| !gapinput@dG:=NonCompactDimension(G); 243| 244 42 245\end{Verbatim} 246 } 247 248 249 250\subsection{\textcolor{Chapter }{PCoefficients}} 251\logpage{[ 2, 1, 2 ]}\nobreak 252\hyperdef{L}{X868FC1B77B4B4D4C}{} 253{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{PCoefficients({\mdseries\slshape type, rank})\index{PCoefficients@\texttt{PCoefficients}} 254\label{PCoefficients} 255}\hfill{\scriptsize (function)}}\\ 256 257 258 Let $G$ be a compact connected Lie group of the type \mbox{\texttt{\mdseries\slshape type}} and the rank \mbox{\texttt{\mdseries\slshape rank}}. Let $\Lambda\,P_{G}=\Lambda (y_1,...,y_l)$ be the exterior algebra over the spaces $P_G$ of the primitive elements in $H^*(G)$. Denote the degrees as follows $|y_j|=2p_j-1,j=1,...,l$. This function returns coefficients $p_1,\ldots,p_l$. 259\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] 260 !gapprompt@gap>| !gapinput@PCoefficients("D",5); 261| 262 [ 2, 4, 6, 8, 5 ] 263\end{Verbatim} 264 } 265 266 267 268\subsection{\textcolor{Chapter }{PCalculate}} 269\logpage{[ 2, 1, 3 ]}\nobreak 270\hyperdef{L}{X827DC41787C8BC7B}{} 271{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{PCalculate({\mdseries\slshape pi, qi})\index{PCalculate@\texttt{PCalculate}} 272\label{PCalculate} 273}\hfill{\scriptsize (function)}}\\ 274 275 276 Here $pi=\{ p_1,\ldots,p_l\}$ and $qi=\{ q_1,\ldots,q_m\}$ are sets of coefficients ($l\geq m$). This function returns the polynomial: $P(t)=\prod_{j=m+1}^l(1+t^{2p_j-1})\prod_{i=1}^m(1-t^{2p_i})/(1-t^{2q_i})$. 277\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] 278 !gapprompt@gap>| !gapinput@PCalculate([4,2,3],[2,2]); 279| 280 t^9+t^5+t^4+1 281\end{Verbatim} 282 } 283 284 285 286\subsection{\textcolor{Chapter }{AllZeroDH}} 287\logpage{[ 2, 1, 4 ]}\nobreak 288\hyperdef{L}{X87164CA87A56E5B8}{} 289{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{AllZeroDH({\mdseries\slshape type, rank, id})\index{AllZeroDH@\texttt{AllZeroDH}} 290\label{AllZeroDH} 291}\hfill{\scriptsize (function)}}\\ 292 293 294 Let $G^C$ be a complex Lie algebra of the type \mbox{\texttt{\mdseries\slshape type}} and the rank \mbox{\texttt{\mdseries\slshape rank}}. Let $G$ be a real form of $G^C$ with the index \mbox{\texttt{\mdseries\slshape id}} (see \mbox{\texttt{\mdseries\slshape RealFormsInformation}},\cite{CoReLG}). This function returns the set of degrees of $P(t)$ that have zero coefficients over all permutation (see Section 7 in \cite{our}). 295\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] 296 !gapprompt@gap>| !gapinput@AllZeroDH("F",4,2); 297| 298 [ 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27 ] 299\end{Verbatim} 300 } 301 302 } 303 304 } 305 306 307\chapter{\textcolor{Chapter }{Algorithm example}}\logpage{[ 3, 0, 0 ]} 308\hyperdef{L}{X7F57F15D7A4099A1}{} 309{ 310 In this chapter we use additionaly functions from the following packages: 311CoReLG \cite{CoReLG} and SLA \cite{SLA}. We will show in detail the split case (for a non-split case you should use 312algoritm to generate regular subalgebras from \cite{DFG}). For example, we take $G=\mathfrak{e}_{6(6)}$ (tuple "E",6,2 in CoReLG notation). We calculate \mbox{\texttt{\mdseries\slshape AllZeroDH}} on it. 313\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] 314 !gapprompt@gap>| !gapinput@AllZeroDH("E",6,2); 315| 316 [ 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 27, 317 28, 29, 30, 31, 32, 35, 36, 37, 38, 39, 40, 41 ] 318\end{Verbatim} 319 We generate all regular subalgebras of complexification. 320\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] 321 !gapprompt@gap>| !gapinput@GC:=SimpleLieAlgebra("E",6,Rationals);; 322| 323 !gapprompt@gap>| !gapinput@REG:=RegularSemisimpleSubalgebras(GC);; 324| 325 !gapprompt@gap>| !gapinput@L0:=List( REG, SemiSimpleType ); 326| 327 [ "A1", "A1 A1", "A2 A1", "A4", "D5", "A4 A1", "A2 A1 A1", "A2 A1 A2", "A3 A1", 328 "A1 A1 A1", "A2", "A3", "A5", "A2 A2", "D4", "A5 A1", "A3 A1 A1", "A1 A1 A1 A1", 329 "A2 A2 A2" ] 330\end{Verbatim} 331 For each subalgebras we take the split real form and calculate its non-compact 332dimension. 333\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] 334 !gapprompt@gap>| !gapinput@L0[4]; 335| 336 "A4" 337 !gapprompt@gap>| !gapinput@RealFormsInformation( "A", 4 ); 338| 339 340 There are 4 simple real forms with complexification A4 341 1 is of type su(5), compact form 342 2 - 3 are of type su(p,5-p) with 1 <= p <= 2 343 4 is of type sl(5,R) 344 Index '0' returns the realification of A4 345 346 !gapprompt@gap>| !gapinput@G:=RealFormById("A",4,4);; 347| 348 !gapprompt@gap>| !gapinput@NonCompactDimension( G ); 349| 350 14 351\end{Verbatim} 352 Number 14 is in output of \mbox{\texttt{\mdseries\slshape AllZeroDH}} function, so for $\mathfrak{g}=e_{6(6)}$ and $\mathfrak{h}=\mathfrak{sl}(5,\mathbb{R})$ corresponding homogeneous spaces $G/H$ do not have compact Clifford{\textendash}Klein forms. 353\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] 354 !gapprompt@gap>| !gapinput@L0[5]; 355| 356 "D5" 357 !gapprompt@gap>| !gapinput@RealFormsInformation( "D", 5 ); 358| 359 360 There are 7 simple real forms with complexification D5 361 1 is of type so(10), compact form 362 2 - 3 are of type so(2p,10-2p) with 1 <= p <= 2 363 4 is of type so*(10) 364 5 is of type so(9,1) 365 6 - 7 are of type so(2p+1,10-2p-1) with 1 <= p <= 2 366 Index '0' returns the realification of D5 367 368 !gapprompt@gap>| !gapinput@G:=RealFormById("D",5,7);; 369| 370 !gapprompt@gap>| !gapinput@NonCompactDimension( G ); 371| 372 25 373\end{Verbatim} 374 Number 25 is not in output of \mbox{\texttt{\mdseries\slshape AllZeroDH}} function, so for $\mathfrak{g}=e_{6(6)}$ and $\mathfrak{h}=\mathfrak{so}(5,5)$ our algoritm does not provide a solution to the problem. } 375 376 \def\bibname{References\logpage{[ "Bib", 0, 0 ]} 377\hyperdef{L}{X7A6F98FD85F02BFE}{} 378} 379 380\bibliographystyle{alpha} 381\bibliography{NoCKbib.xml} 382 383\addcontentsline{toc}{chapter}{References} 384 385\def\indexname{Index\logpage{[ "Ind", 0, 0 ]} 386\hyperdef{L}{X83A0356F839C696F}{} 387} 388 389\cleardoublepage 390\phantomsection 391\addcontentsline{toc}{chapter}{Index} 392 393 394\printindex 395 396\newpage 397\immediate\write\pagenrlog{["End"], \arabic{page}];} 398\immediate\closeout\pagenrlog 399\end{document} 400